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I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction.

reference frame 1: source travelling towards x-direction. observer at rest.

Δto=1/fo - observer's time between subsequent wave crests.

due to time dilation, source's time should be running slower Δ[itex]\tau[/itex]s=Δto/[itex]\gamma[/itex], so actual frequency of source is 1/Δ[itex]\tau[/itex]s=

**fs=fo*[itex]\gamma[/itex]**(the correct relation)

reference frame 2: observer travelling towards x-direction. source at rest.

Δts=1/fs - source's time between subsequent wave crests.

due to time dilation, observer's time should be running slower Δ[itex]\tau[/itex]o=Δts/[itex]\gamma[/itex], so frequency observed is 1/Δ[itex]\tau[/itex]o=fo=[itex]\gamma[/itex]fs

which leaves actual frequenct of source

**fs=fo/[itex]\gamma[/itex]**

why are these two results seemingly inconsistent?

thanks in advance.